Hodge theory

Hodge cycle

In differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. A homology class x in a homology group where V is a non-singular complex algebraic variety or Kähler manifold is a Hodge cycle, provided it satisfies two conditions. Firstly, k is an even integer , and in the direct sum decomposition of H shown to exist in Hodge theory, x is purely of type . Secondly, x is a rational class, in the sense that it lies in the image of the abelian group homomorphism defined in algebraic topology (as a special case of the universal coefficient theorem). The conventional term Hodge cycle therefore is slightly inaccurate, in that x is considered as a class (modulo boundaries); but this is normal usage. The importance of Hodge cycles lies primarily in the Hodge conjecture, to the effect that Hodge cycles should always be algebraic cycles, for V a complete algebraic variety. This is an unsolved problem, as of March 2022; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known. (Wikipedia).

Video thumbnail

C07 Homogeneous linear differential equations with constant coefficients

An explanation of the method that will be used to solve for higher-order, linear, homogeneous ODE's with constant coefficients. Using the auxiliary equation and its roots.

From playlist Differential Equations

Video thumbnail

What is the parent function of a linear graph

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

What is a linear equation

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

What are the x and y intercepts of a linear equation

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

C34 Expanding this method to higher order linear differential equations

I this video I expand the method of the variation of parameters to higher-order (higher than two), linear ODE's.

From playlist Differential Equations

Video thumbnail

Hodge theory and algebraic cycles - Phillip Griffiths

Geometry and Arithmetic: 61st Birthday of Pierre Deligne Phillip Griffiths Institute for Advanced Study October 18, 2005 Pierre Deligne, Professor Emeritus, School of Mathematics. On the occasion of the sixty-first birthday of Pierre Deligne, the School of Mathematics will be hosting a f

From playlist Pierre Deligne 61st Birthday

Video thumbnail

Hodge Theory -- From Abel to Deligne - Phillip Griffiths

Phillip Griffiths School of Mathematics, Institute for Advanced Study October 14, 2013 For more videos, visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

How do you graph an equation using the intercept method

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

What is everything you need to know to graph an equation in slope intercept form

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

Some algebro-geometric aspects of limiting mixed Hodge structure - Phillip Griffiths

Phillip Griffiths Professor Emeritus, School of Mathematics December 16, 2014 This will be an expository talk, mostly drawn from the literature and with emphasis on the several parameter case of degenerating families of algebraic varieties. More videos on http://video.ias.edu

From playlist Mathematics

Video thumbnail

Christian Schnell

https://www.math.ias.edu/files/media/agenda.pdf More videos on http://video.ias.edu

From playlist Mathematics

Video thumbnail

Matt Kerr

https://www.math.ias.edu/files/media/agenda.pdf More videos on http://video.ias.edu

From playlist Mathematics

Video thumbnail

The Hodge conjecture for moduli spaces of stable sheaves over a nodal curve by Inder Kaur

DISCUSSION MEETING: MODULI OF BUNDLES AND RELATED STRUCTURES ORGANIZERS : Rukmini Dey and Pranav Pandit DATE: 10 February 2020 to 14 February 2020 VENUE: Ramanujan Lecture Hall, ICTS, Bangalore Background: At its core, much of mathematics is concerned with the problem of classifying

From playlist Moduli Of Bundles And Related Structures 2020

Video thumbnail

What is the slope of a linear equation

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

Geometers Abandoned 2,000 Year-Old Math. This Million-Dollar Problem was Born - Hodge Conjecture

The Hodge Conjecture is one of the deepest problems in analytic geometry and one of the seven Millennium Prize Problems worth a million dollars, offered by the Clay Mathematical Institute in 2000. It consists of drawing shapes known topological cycles on special surfaces called projective

From playlist Math

Video thumbnail

Spencer Bloch 1/3 - Mixed Motives [1991]

Michio Kuga Memorial Lecture Series Stony Brook University Department of Mathematics and Institute for Mathematical Sciences Spencer Bloch (University of Chicago) Mixed Motives Lectures March 1991 http://www.math.stonybrook.edu/Videos/Kuga/Bloch-1991/

From playlist Number Theory

Video thumbnail

Hodge theory, between algebraicity and transcendence (Lecture 2) by Bruno Klingler

DISCUSSION MEETING TOPICS IN HODGE THEORY (HYBRID) ORGANIZERS: Indranil Biswas (TIFR, Mumbai, India) and Mahan Mj (TIFR, Mumbai, India) DATE: 20 February 2023 to 25 February 2023 VENUE: Ramanujan Lecture Hall and Online This is a followup discussion meeting on complex and algebraic ge

From playlist Topics in Hodge Theory - 2023

Video thumbnail

Summary for graph an equation in Standard form

👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu

From playlist ⚡️Graph Linear Equations | Learn About

Video thumbnail

Mumford-Tate Groups and Domains - Phillip Griffiths

Phillip Griffiths Professor Emeritus, School of Mathematics March 28, 2011 For more videos, visit http://video.ias.edu

From playlist Mathematics

Related pages

Hodge theory | Kähler manifold | Complex number | Differential geometry | Direct sum of modules | Algebraic variety | Hodge conjecture | Universal coefficient theorem | Algebraic cycle | Algebraic topology